Why does the entangled pair (the mechanism of teleportation) need to entangle to start with in order for the teleportation to proceed? After you perform the Bell measurement, doesn't it just break the entanglement anyway?
You can think of entanglement as a way to share the information between several qubits (this is definition of entanglement - the state of the system in which information is distributed across the whole system and can not be represented as states of individual subsystems).
- First Alice and Bob's parts of Bell pair are entangled.
- Then Alice's message qubit is entangled with Alice's part of Bell pair, so that information carried by it (the $\alpha$ and $\beta$ in the $\alpha |0\rangle + \beta |1\rangle$ state) is distributed across the whole system.
- Measuring two qubits in Alice's possession "squeezes" the information out of them and into the last part of the system, Bob's qubit.
Without the initial entanglement, Alice's qubits don't have a way to share information with Bob's qubit.
In more mathematical terms,
- If Alice's and Bob's qubits are not entangled initially, the state of the system can be represented as tensor product $|\psi_A\rangle\otimes|\psi_B\rangle$.
- Adding message qubit corresponds to the state $|m_A\rangle \otimes |\psi_A\rangle\otimes|\psi_B\rangle$
- Now, if you only perform gates on the first two qubits, you can always represent them as $G_A\otimes I_B$, and the state of Bob's qubit never changes.